Does not cooking rice in enough water so that it completely absorbed when stopped cooking contain higher levels of arsenic if the rice is overcooked?

Unlikely: It is an issue as arsenic can accumulate in rice. The main precaution recommended is pre-soaking and cooking in lots of water 5:1. It is more likely to be an issue if rice is the staple (main) food. See below: http://www.bbc.com/news/health-38910848 For specific advice suggest discussion with doctor or nutritionist licensed in your area.

Best plan: Try to rinse raw rice in strainer several times till water runs nearly clear. That rids the surface of starch and most of the possible bad things. Also (sadly) brown (healthier) rice may have greater arsenic risk. Also, cook it as you know to be the right way. I don't think over/under increases the arsenic - rinsing does help.